Activity Gibbs-Duhem integration

In theory, once the activity of an electrolyte in solution is known, the activity of the solvent can be determined by the Gibbs-Duhem integration (see section 2.11). In practice, the calculation is prohibitive, because of the chemical complexity of most aqueous solutions of geochemical interest. Semiempirical approximations are therefore preferred, such as that proposed by Helgeson (1969), consisting of a simulation of the properties of the H20-NaCl system up to a solute... 

Figure 8.7 Estimation of activities of MnO in FeO-MnO system by Gibbs Duhem integration...

The partial pressures of Cu in the system Cu-Fe-Pt in the temperature range 1240 to 1360°C have been measured by the Knudsen effusion technique and the thermodynamic properties of this system at 1300°C have been derived [1989Par]. The activities of Fe in solid solutions at 1300°C were calculated by Gibbs-Duhem integration of the Cu activities. The experimental alloys were prepared from Cu (99.999 mass%), Fe (99.999 mass%) and Pt (99.99 mass%) by induction melting in an alumina cmcible under an Ar atmosphere. The alloy buttons were then homogenized in a H2 atmosphere for 5 to 30 days at 900 to 1300°C. 

A similar situation exists for alloys where a component pressure is not measurable under the temperatures for measurable vapor pressures of other components. Examples are Hf in Ni-Al-Hf [85]), Cr in Mn-Cr [104], and rare earth (RE) in Mg-RE alloys [105]. The latter study was based on a Knudsen cell without the use of a mass spectrometer. Nonetheless the approach is applicable to a mass spectrometric study. Depending on the alloy system, several approaches can be taken. In some cases the effect of controlled additions of the low-pressure component on a fixed ratio of the measured elements is the only required information [100,102]. Albers et al. [85] and Zaitsev et al. [104] did a Gibbs-Duhem integration to obtain the activities of the low vapor pressure component. Pahlman and Smith [105] assumed Raoultian behavior in the terminal RE-Mg solution and moved across the phase diagram to derive the activity of the RE component in each two-phase region. 

The system evaporates UC2(g) as well as a small amount of UC4(g). Since the UC2(g)/U,g) pressure ratio is proportional to the carbon activity squared. Storms (1966a) was able to determine the carbon activity directly by measuring this ratio at various temperatures and compositions. Thus, the carbon activity was obtained without measuring any pure carbon specie. The result is compared to the uranium activity in Fig. 69 and Table 74. A curve for the carbon activity was calculated by a Gibbs-Duhem integration of the uranium activity curve and the result is compared to the direct measurements in the figure. 

Find the activity coefficient of calcium chloride at 1.000 mol kg, using a Gibbs-Duhem integration. 

Equation (16) is a differential equation and applies equally to activity coefficients normalized by the symmetric or unsymme-tric convention. It is only in the integrated form of the Gibbs-Duhem equation that the type of normalization enters as a boundary condition. 

If we vary the composition of a liquid mixture over all possible composition values at constant temperature, the equilibrium pressure does not remain constant. Therefore, if integrated forms of the Gibbs-Duhem equation [Equation (16)] are used to correlate isothermal activity coefficient data, it is necessary that all activity coefficients be evaluated at the same pressure. Unfortunately, however, experimentally obtained isothermal activity coefficients are not all at the same pressure and therefore they must be corrected from the experimental total pressure P to the same (arbitrary) reference pressure designated P. This may be done by the rigorous thermodynamic relation at constant temperature and composition ... 

The behaviour of most metallurgically important solutions could be described by certain simple laws. These laws and several other pertinent aspects of solution behaviour are described in this section. The laws of Raoult, Henry and Sievert are presented first. Next, certain parameters such as activity, activity coefficient, chemical potential, and relative partial and integral molar free energies, which are essential for thermodynamic detailing of solution behaviour, are defined. This is followed by a discussion on the Gibbs-Duhem equation and ideal and nonideal solutions. The special case of nonideal solutions, termed as a regular solution, is then presented wherein the concept of excess thermodynamic functions has been used. 

A graphical integration of the Gibbs-Duhem equation is not necessary if an analytical expression for the partial properties of mixing is known. Let us assume that we have a dilute solution that can be described using the activity coefficient at infinite dilution and the self-interaction coefficients introduced in eq. (3.64). 

For the ternary solution, the Gibbs-Duhem equation can be easily integrated to calculate the activity coefficient of water when the expressions for the activity coefficients of the electrolytes are written at constant molality. For Harned s rule, integration of the Gibbs-Duhem equation gives the activity of water as ... 

An alternative approach is to estimate activity coefficients of the solvents from experimental data and correlate these coefficients using, for example, the Wilson equation. Rousseau et al. (3) and Jaques and Furter (4) have used the Wilson equation, as well as other integrated forms of the Gibbs-Duhem equation, to show the utility of this approach. These authors found it necessary, however, to modify the definitions of the solvent reference states so that the results could be normalized. 

Two limitations are involved in the derivation of the above equation (1) the compositions of mixed solvents (points c and d) should be close enough to each other for the trapezoidal mle used to integrate the Gibbs-Duhem equation to be valid, (2) the solubility of the solid should be low enough for the activity coefficients of the solvent and cosolvent to be taken equal to those in a solute-free binary solvent mixture. In addition, the fugacity of the solid phase in Eq. (4) should remain the same for all mixed solvent compositions considered. 

By using activity coefficients obtained from b.m.f. measurements, integration of the Gibbs-Duhem equation permits the evaluation of the activity of the solvent (water) cf., Newton and Tippetts, J. Am. Chem. Soc., 58, 280 (1936). 

Determination of the activity coefficients of the non-volatile solute in a solution is difficult. If electrolytes (ions) are present, the activities can be obtained from experimental electromotive force (EMF) measurements. However, for non-electrolyte and non-volatile solutes an indirect method is applied to find initially the activity of the solvent over a range of solute concentrations, and then the Gibbs-Duhem equation is integrated to find the solute activity. If the solution is saturated, then it is easy to calculate the activity coefficient... 

These data can be studied in two ways. The first is to use the Gibbs-Duhem equation and numerical integration methods to calculate the vapor-phase mole fractions, as considered in Problem 10.2-6. A second method is to choose a liquid-phase activity coefficient model and determine the values of the parameters in the model that give the best fit of the experimental data. We have, from Eq. 10.2-2b, that at the jth experimental point... 

The isopiestic method has proved invaluable at molalities down to 0.1 but below this it becomes too inaccurate. Consequently activity coefficients at 0.1 moledity have to be estimated by some such method as used in this problem. Activity coefficients at higher molalities can then be obtained by integrating the Gibbs-Duhem relation. A comprehensive survey of results thus obtained is given by Robinson and Stokes (Trans. Faraday Soc. 1949, 45, 612). 

Mean activity coefficients derived from cryoscopic measurements have been tabulated by C. M. Criss in Appendices 2.4.15-19. These values should be used with caution as considerable uncertainties can arise from (a) the experimental method, fb) the value used for the cryoscopic constant, (c) integration using the Gibbs-Duhem equation (eqn. 2.8.13). 

Moderate temperature changes result in such minor changes in activity coefficient that constant-pressure data are ordinarily satisfactory for application of the various integrated forms of the Gibbs-Duhem equation. 

Over the range of concentrations between the solubility limits the apparent activity coefficients will vary inversely as the concentrations based on the mixture as a whole. Elimination of 7 s between Eq. (3.79) and any of the integrated Gibbs-Duhem equations therefore permits the estimation of Aab and Aba from the mutual solubility. Thus, the Margules equations lead to... 

Calculate activity coefficients for the system chloroform-acetone at 35.17 C. from vapor-liquid data reported in International Critical Tables (Vol. Ill, p. 286). Fit one of the integrated Gibbs-Duhem equations to the data. With the help of heat of solution data, ibid. Vol. V, pp. 151, 155, 158, estimate the values of the equation constants for 55.1 C., and calculate the activity coefficients for this temperature. Compare with those computed from vapor-liquid data at this temperature, ihid.j Vol. Ill, p. 286. 

Calculate the activity coefficients from azeotropic data for the following systems using one of the integrated Gibbs-Duhem equations, obtaining the necessary data from the compilation of Horsley [Ind. Eng, Chem. Anal, Ed, 19, 508 (1947)]. Compare with those calculated from the complete vapor-liquid data, as reported in Chemical Engineers Handbook. ... 

If the logarithms of the activity coefficients of a binary system are plotted against molar composition, Eq. (53) relates the slopes at any value of the x s. This is useful for testing the consistency of data. Such testing becomes easier if we have a systematic way to correlate and smooth data and extend them over the entire range of composition. Integrated forms of the Gibbs-Duhem relation allow us to do this. Theoretically, one reliable measurement of vapor-liquid equilibrium at any point can then be used to characterize a system. 

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